2007 NIPS Tutorial on:Deep Belief Nets Geoffrey Hinton Canadian Institute for Advanced Research & Department of Computer Science University of Toronto
Some things you will learn in this tutorial • How to learn multi-layer generative models of unlabelled data by learning one layer of features at a time. • How to add Markov Random Fields in each hidden layer. • How to use generative models to make discriminative training methods work much better for classification and regression. • How to extend this approach to Gaussian Processes and how to learn complex, domain-specific kernels for a Gaussian Process. • How to perform non-linear dimensionality reduction on very large datasets • How to learn binary, low-dimensional codes and how to use them for very fast document retrieval. • How to learn multilayer generative models of high-dimensional sequential data.
Low-dimensional data (e.g. less than 100 dimensions) Lots of noise in the data There is not much structure in the data, and what structure there is, can be represented by a fairly simple model. The main problem is distinguishing true structure from noise. High-dimensional data (e.g. more than 100 dimensions) The noise is not sufficient to obscure the structure in the data if we process it right. There is a huge amount of structure in the data, but the structure is too complicated to be represented by a simple model. The main problem is figuring out a way to represent the complicated structure so that it can be learned. A spectrum of machine learning tasks Typical Statistics------------Artificial Intelligence
Perceptrons (~1960) used a layer of hand-coded features and tried to recognize objects by learning how to weight these features. There was a neat learning algorithm for adjusting the weights. But perceptrons are fundamentally limited in what they can learn to do. Historical background:First generation neural networks Toy Bomb output units e.g. class labels non-adaptive hand-coded features input units e.g. pixels Sketch of a typical perceptron from the 1960’s
Second generation neural networks (~1985) Compare outputs with correct answer to get error signal Back-propagate error signal to get derivatives for learning outputs hidden layers input vector
A temporary digression • Vapnik and his co-workers developed a very clever type of perceptron called a Support Vector Machine. • Instead of hand-coding the layer of non-adaptive features, each training example is used to create a new feature using a fixed recipe. • The feature computes how similar a test example is to that training example. • Then a clever optimization technique is used to select the best subset of the features and to decide how to weight each feature when classifying a test case. • But its just a perceptron and has all the same limitations. • In the 1990’s, many researchers abandoned neural networks with multiple adaptive hidden layers because Support Vector Machines worked better.
What is wrong with back-propagation? • It requires labeled training data. • Almost all data is unlabeled. • The learning time does not scale well • It is very slow in networks with multiple hidden layers. • It can get stuck in poor local optima.
Overcoming the limitations of back-propagation • Keep the efficiency and simplicity of using a gradient method for adjusting the weights, but use it for modeling the structure of the sensory input. • Adjust the weights to maximize the probability that a generative model would have produced the sensory input. • Learn p(image) not p(label | image) • If you want to do computer vision, first learn computer graphics • What kind of generative model should we learn?
A belief net is a directed acyclic graph composed of stochastic variables. We get to observe some of the variables and we would like to solve two problems: The inference problem: Infer the states of the unobserved variables. The learning problem: Adjust the interactions between variables to make the network more likely to generate the observed data. Belief Nets stochastic hidden cause visible effect We will use nets composed of layers of stochastic binary variables with weighted connections. Later, we will generalize to other types of variable.
These have a state of 1 or 0. The probability of turning on is determined by the weighted input from other units (plus a bias) Stochastic binary units(Bernoulli variables) 1 0 0
It is easy to generate an unbiased example at the leaf nodes, so we can see what kinds of data the network believes in. It is hard to infer the posterior distribution over all possible configurations of hidden causes. It is hard to even get a sample from the posterior. So how can we learn deep belief nets that have millions of parameters? Learning Deep Belief Nets stochastic hidden cause visible effect
The learning rule for sigmoid belief nets • Learning is easy if we can get an unbiased sample from the posterior distribution over hidden states given the observed data. • For each unit, maximize the log probability that its binary state in the sample from the posterior would be generated by the sampled binary states of its parents. j i learning rate
Explaining away (Judea Pearl) • Even if two hidden causes are independent, they can become dependent when we observe an effect that they can both influence. • If we learn that there was an earthquake it reduces the probability that the house jumped because of a truck. -10 -10 truck hits house earthquake posterior 20 20 p(1,1)=.0001 p(1,0)=.4999 p(0,1)=.4999 p(0,0)=.0001 -20 house jumps
To learn W, we need the posterior distribution in the first hidden layer. Problem 1: The posterior is typically complicated because of “explaining away”. Problem 2: The posterior depends on the prior as well as the likelihood. So to learn W, we need to know the weights in higher layers, even if we are only approximating the posterior. All the weights interact. Problem 3: We need to integrate over all possible configurations of the higher variables to get the prior for first hidden layer. Yuk! Why it is usually very hard to learn sigmoid belief nets one layer at a time hidden variables hidden variables prior hidden variables likelihood W data
Two types of generative neural network • If we connect binary stochastic neurons in a directed acyclic graph we get a Sigmoid Belief Net (Radford Neal 1992). • If we connect binary stochastic neurons using symmetric connections we get a Boltzmann Machine (Hinton & Sejnowski, 1983). • If we restrict the connectivity in a special way, it is easy to learn a Boltzmann machine.
Restricted Boltzmann Machines(Smolensky ,1986, called them “harmoniums”) hidden • We restrict the connectivity to make learning easier. • Only one layer of hidden units. • We will deal with more layers later • No connections between hidden units. • In an RBM, the hidden units are conditionally independent given the visible states. • So we can quickly get an unbiased sample from the posterior distribution when given a data-vector. • This is a big advantage over directed belief nets j i visible
The Energy of a joint configuration(ignoring terms to do with biases) binary state of visible unit i binary state of hidden unit j Energy with configuration v on the visible units and h on the hidden units weight between units i and j
Weights Energies Probabilities • Each possible joint configuration of the visible and hidden units has an energy • The energy is determined by the weights and biases (as in a Hopfield net). • The energy of a joint configuration of the visible and hidden units determines its probability: • The probability of a configuration over the visible units is found by summing the probabilities of all the joint configurations that contain it.
The probability of a joint configuration over both visible and hidden units depends on the energy of that joint configuration compared with the energy of all other joint configurations. The probability of a configuration of the visible units is the sum of the probabilities of all the joint configurations that contain it. Using energies to define probabilities partition function
A picture of the maximum likelihood learning algorithm for an RBM j j j j a fantasy i i i i t = 0 t = 1 t = 2 t = infinity Start with a training vector on the visible units. Then alternate between updating all the hidden units in parallel and updating all the visible units in parallel.
A quick way to learn an RBM Start with a training vector on the visible units. Update all the hidden units in parallel Update the all the visible units in parallel to get a “reconstruction”. Update the hidden units again. j j i i t = 0 t = 1 reconstruction data This is not following the gradient of the log likelihood. But it works well. It is approximately following the gradient of another objective function (Carreira-Perpinan & Hinton, 2005).
How to learn a set of features that are good for reconstructing images of the digit 2 50 binary feature neurons 50 binary feature neurons Decrement weights between an active pixel and an active feature Increment weights between an active pixel and an active feature 16 x 16 pixel image 16 x 16 pixel image data (reality) reconstruction (better than reality)
The final 50 x 256 weights Each neuron grabs a different feature.
How well can we reconstruct the digit images from the binary feature activations? Reconstruction from activated binary features Reconstruction from activated binary features Data Data New test images from the digit class that the model was trained on Images from an unfamiliar digit class (the network tries to see every image as a 2)
Three ways to combine probability density models (an underlying theme of the tutorial) • Mixture: Take a weighted average of the distributions. • It can never be sharper than the individual distributions. It’s a very weak way to combine models. • Product: Multiply the distributions at each point and then renormalize. • Exponentially more powerful than a mixture. The normalization makes maximum likelihood learning difficult, but approximations allow us to learn anyway. • Composition: Use the values of the latent variables of one model as the data for the next model. • Works well for learning multiple layers of representation, but only if the individual models are undirected.
Training a deep network(the main reason RBM’s are interesting) • First train a layer of features that receive input directly from the pixels. • Then treat the activations of the trained features as if they were pixels and learn features of features in a second hidden layer. • It can be proved that each time we add another layer of features we improve a variational lower bound on the log probability of the training data. • The proof is slightly complicated. • But it is based on a neat equivalence between an RBM and a deep directed model (described later)
To generate data: Get an equilibrium sample from the top-level RBM by performing alternating Gibbs sampling for a long time. Perform a top-down pass to get states for all the other layers. So the lower level bottom-up connections are not part of the generative model. They are just used for inference. The generative model after learning 3 layers h3 h2 h1 data
Why does greedy learning work? An aside: Averaging factorial distributions • If you average some factorial distributions, you do NOT get a factorial distribution. • In an RBM, the posterior over the hidden units is factorial for each visible vector. • But the aggregated posterior over all training cases is not factorial (even if the data was generated by the RBM itself).
Each RBM converts its data distribution into an aggregated posterior distribution over its hidden units. This divides the task of modeling its data into two tasks: Task 1: Learn generative weights that can convert the aggregated posterior distribution over the hidden units back into the data distribution. Task 2: Learn to model the aggregated posterior distribution over the hidden units. The RBM does a good job of task 1 and a moderately good job of task 2. Task 2 is easier (for the next RBM) than modeling the original data because the aggregated posterior distribution is closer to a distribution that an RBM can model perfectly. Why does greedy learning work? Task 2 aggregated posterior distribution on hidden units Task 1 data distribution on visible units
Why does greedy learning work? The weights, W, in the bottom level RBM define p(v|h) and they also, indirectly, define p(h). So we can express the RBM model as If we leave p(v|h) alone and improve p(h), we will improve p(v). To improve p(h), we need it to be a better model of the aggregated posterior distribution over hidden vectors produced by applying W to the data.
Which distributions are factorial in a directed belief net? • In a directed belief net with one hidden layer, the posterior over the hidden units for each visible vector is non-factorial (due to explaining away). • The aggregated posterior is factorial if the data was generated by the directed model. • It’s the opposite way round from an undirected model. • The intuitions that people have from using directed models are very misleading for undirected models.
A directed module also converts its data distribution into an aggregated posterior Task 1 is now harder because the posterior for each training case is non-factorial. Task 2 is performed using an independent prior. This is a bad approximation unless the aggregated posterior is close to factorial. A directed module attempts to make the aggregated posterior factorial in one step. This is too difficult and leads to a bad compromise. There is no guarantee that the aggregated posterior is easier to model than the data distribution. Why does greedy learning fail in a directed module? Task 2 aggregated posterior distribution on hidden units Task 1 data distribution on visible units
A model of digit recognition The top two layers form an associative memory whose energy landscape models the low dimensional manifolds of the digits. The energy valleys have names 2000 top-level neurons 10 label neurons 500 neurons The model learns to generate combinations of labels and images. To perform recognition we start with a neutral state of the label units and do an up-pass from the image followed by a few iterations of the top-level associative memory. 500 neurons 28 x 28 pixel image
Fine-tuning with a contrastive version of the “wake-sleep” algorithm After learning many layers of features, we can fine-tune the features to improve generation. 1. Do a stochastic bottom-up pass • Adjust the top-down weights to be good at reconstructing the feature activities in the layer below. • Do a few iterations of sampling in the top level RBM -- Adjust the weights in the top-level RBM. • Do a stochastic top-down pass • Adjust the bottom-up weights to be good at reconstructing the feature activities in the layer above.
Show the movie of the network generating digits (available at www.cs.toronto/~hinton)
Samples generated by letting the associative memory run with one label clamped. There are 1000 iterations of alternating Gibbs sampling between samples.
Examples of correctly recognized handwritten digitsthat the neural network had never seen before Its very good
How well does it discriminate on MNIST test set with no extra information about geometric distortions? • Generative model based on RBM’s 1.25% • Support Vector Machine (Decoste et. al.) 1.4% • Backprop with 1000 hiddens (Platt) ~1.6% • Backprop with 500 -->300 hiddens ~1.6% • K-Nearest Neighbor ~ 3.3% • See Le Cun et. al. 1998 for more results • Its better than backprop and much more neurally plausible because the neurons only need to send one kind of signal, and the teacher can be another sensory input.
Unsupervised “pre-training” also helps for models that have more data and better priors • Ranzato et. al. (NIPS 2006) used an additional 600,000 distorted digits. • They also used convolutional multilayer neural networks that have some built-in, local translational invariance. Back-propagation alone: 0.49% Unsupervised layer-by-layer pre-training followed by backprop: 0.39% (record)
Another view of why layer-by-layer learning works • There is an unexpected equivalence between RBM’s and directed networks with many layers that all use the same weights. • This equivalence also gives insight into why contrastive divergence learning works.
The distribution generated by this infinite directed net with replicated weights is the equilibrium distribution for a compatible pair of conditional distributions: p(v|h) and p(h|v) that are both defined by W A top-down pass of the directed net is exactly equivalent to letting a Restricted Boltzmann Machine settle to equilibrium. So this infinite directed net defines the same distribution as an RBM. An infinite sigmoid belief net that is equivalent to an RBM etc. h2 v2 h1 v1 h0 v0
The variables in h0 are conditionally independent given v0. Inference is trivial. We just multiply v0 by W transpose. The model above h0 implements a complementary prior. Multiplying v0 by W transpose gives the product of the likelihood term and the prior term. Inference in the directed net is exactly equivalent to letting a Restricted Boltzmann Machine settle to equilibrium starting at the data. Inference in a directed net with replicated weights etc. h2 v2 h1 v1 + + h0 + + v0
The learning rule for a sigmoid belief net is: With replicated weights this becomes: etc. h2 v2 h1 v1 h0 v0
First learn with all the weights tied This is exactly equivalent to learning an RBM Contrastive divergence learning is equivalent to ignoring the small derivatives contributed by the tied weights between deeper layers. Learning a deep directed network etc. h2 v2 h1 v1 h0 h0 v0 v0
Then freeze the first layer of weights in both directions and learn the remaining weights (still tied together). This is equivalent to learning another RBM, using the aggregated posterior distribution of h0 as the data. etc. h2 v2 h1 v1 v1 h0 h0 v0
How many layers should we use and how wide should they be? (I am indebted to Karl Rove for this slide) • How many lines of code should an AI program use and how long should each line be? • This is obviously a silly question. • Deep belief nets give the creator a lot of freedom. • How best to make use of that freedom depends on the task. • With enough narrow layers we can model any distribution over binary vectors (Sutskever & Hinton, 2007) • If freedom scares you, stick to convex optimization of shallow models that are obviously inadequate for doing Artificial Intelligence.
What happens when the weights in higher layers become different from the weights in the first layer? • The higher layers no longer implement a complementary prior. • So performing inference using the frozen weights in the first layer is no longer correct. • Using this incorrect inference procedure gives a variational lower bound on the log probability of the data. • We lose by the slackness of the bound. • The higher layers learn a prior that is closer to the aggregated posterior distribution of the first hidden layer. • This improves the network’s model of the data. • Hinton, Osindero and Teh (2006) prove that this improvement is always bigger than the loss.
Each RBM has the same subscript as its hidden layer. Each RBM defines its own distribution over its visible vectors Each RBM defines its own distribution over its hidden vectors A stack of RBM’s(Yee-Whye Teh’s idea) hL h2 h1 v
The variational bound Each time we replace the prior over the hidden units by a better prior, we win by the difference in the probability assigned Now we cancel out all of the partition functions except the top one and replace log probabilities by goodnesses using the fact that: This has simple derivatives that give a more justifiable fine-tuning algorithm than contrastive wake-sleep.
Summary so far • Restricted Boltzmann Machines provide a simple way to learn a layer of features without any supervision. • Maximum likelihood learning is computationally expensive because of the normalization term, but contrastive divergence learning is fast and usually works well. • Many layers of representation can be learned by treating the hidden states of one RBM as the visible data for training the next RBM (a composition of experts). • This creates good generative models that can then be fine-tuned. • Contrastive wake-sleep can fine-tune generation.